Inverse Square Law Calculator

Double the distance from a light, a speaker or a radiation source and the intensity does not halve, it quarters. This inverse square law calculator starts from a known intensity at a known distance and solves for the intensity at a new distance, or for the distance that produces a target intensity, printing both ratios alongside so you can see the relationship, not just the answer.

Quick Answer

How to use the inverse square law calculator

Always enter the initial intensity and its distance first; those two anchor everything. Then leave exactly one of the final fields blank, and the blank one is what the tool solves for.

1. Intensity at a new distance: enter I₁, d₁ and d₂, leave I₂ blank. The pre-filled example does this: 1000 at 2, checked at 4, returns 250.000000.
2. Distance for a target intensity: enter I₁, d₁ and the I₂ you want, leave d₂ blank. Enter 100, 1 and 25 and the tool returns a final distance of 2.000000.
3. Checking a known pair: fill in all four fields and the tool simply reports the two ratios. It does not test whether your pair actually obeys the law, so a mismatched pair passes silently (see the limits section below).
4. No target yet: leave both final fields blank and the tool works the doubling example for your own I₁ and d₁, labelling the outputs as an example so they cannot be mistaken for a solved answer.

The inverse square law formula, and where the square comes from

The working relation is I₁/I₂ = (d₂/d₁)². Rearranged to solve for intensity it reads I₂ = I₁ × (d₁/d₂)², and rearranged to solve for distance it reads d₂ = d₁ × √(I₁/I₂), which are the two forms the calculator switches between depending on which field you leave blank.

The square is geometry, not convention. A point source radiates evenly over the surface of an expanding sphere, and that surface grows as 4πr², the sphere area in the SI Brochure’s system of quantities. The same total power spread over four times the area at twice the radius leaves one quarter of the intensity at each point. The gravitational and electric cases follow the same pattern with their own constants, values maintained in the NIST CODATA fundamental constants listing. Sound intensity in free field is treated the same way in OpenStax University Physics, section 17.3 on sound intensity, and gravity’s version is derived in section 13.1 on Newton’s law of universal gravitation.

Worked examples: a stage light, a safe distance, and sunlight at Mars

A stage light reads 1000 lux at 2 metres and the designer needs the level at 4 metres. Enter I₁ = 1000, d₁ = 2, d₂ = 4 with I₂ blank: the tool returns 250.000000 lux, an intensity ratio of 4.000000 and a distance ratio of 2.000000. Run the chart above and the falloff keeps going: 111.111111 lux at 6 metres, 62.500000 at 8 metres.

The distance direction answers the radiation-safety version of the question. A source reads 100 units at 1 metre and the limit is 25: enter I₁ = 100, d₁ = 1, I₂ = 25 with d₂ blank and the tool returns 2.000000 metres, the classic double-the-distance result. For sound work the same geometry sits underneath the decibel version on the decibel distance calculator, where each doubling costs about 6 dB.

Astronomy stretches the same formula over absurd distances. Sunlight at the top of Earth’s atmosphere is about 1361 W/m² at 1 AU; Mars orbits at 1.53 AU. Enter 1361, 1 and 1.53 and the tool returns 581.400316 W/m², matching the commonly published figure of roughly 581 W/m². The stellar-brightness version of this reasoning is what the distance modulus calculator formalises, and the radio-frequency version appears in the EMF distance calculator. Grow-light placement is the same trade in miniature, covered by the grow light distance calculator.

Distance multipliers at a glance

Because the ratios are all that matter, one small table covers every unit system and every application of the law:

Limits of this calculator, stated plainly

Three behaviours of this tool deserve daylight, because each one can produce a confident-looking wrong answer if you miss it. First, the tool only validates the two initial fields. Zero or negative intensities and distances are accepted and flow straight into the arithmetic, where they can produce Infinity or NaN in the output fields; treat either as a sign an input is wrong, not as a result. Second, when both final fields are filled in, the tool reports the ratios for the pair you gave without checking that the pair obeys the law, so it cannot be used to verify a measurement. Third, the physics itself is an idealisation: it assumes a point source, open space and no absorption, reflection or shielding. Close to a large fixture the point-source assumption fails, and indoors, reflected light adds back some of what geometry took away.

What competing calculators miss

Inverse square tools divide by how much machinery they put between you and a very short formula:

• Omni Calculator handles the generic case elegantly, leaves units open and adds a scientific notation converter, with a solar-irradiance example that checks out at 581.4 W/m² for Mars. It solves one direction per run, though, and shows no ratio outputs, so the relationship itself stays invisible.
• CalculatorsHub solves for any of the four unknowns, including back-solving the original intensity, which is broader than this page. Its reference-pair framing is less direct when you already hold a trusted I₁ and d₁ and just want the falloff and both ratios in one pass.
• FIRGELLI adds a drag-slider visualiser and a source-power mode built on I = P/(4πr²), genuinely useful for engineers who know total power. The mode dropdown and the extra source-power framing are more machinery than a lighting or safety check usually needs.
• Testbook wraps the tool in exam-prep teaching with long worked examples, including a gravity example that answers a different formula (F = Gm₁m₂/d²) inside an inverse square article, which blurs what the calculator itself computes.
• The gap this page fills: no mode dropdown at all. The blank field is the mode. Both ratios print on every run, the neither-blank case demonstrates the doubling behaviour on your own numbers instead of erroring, and the limits above are stated on the page rather than discovered by accident.

Frequently asked questions

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